Sull’equazione det Du = f senza ipotesi di segno

Abstract

We consider the nonlinear problem det?u (x) = f (x) x ? u (x) = x x ? ? where k ? 1 is an integer, is a bounded smooth domain in Rn and f ? Ck ???? satisfies Zf (x) dx = meas. The positivity of f is a standard assumption in the literature. In a recent joint paper with B.Dacorogna and O.Kneuss (EPFL) we prove the existence of a solution u ? Ck ???? ;Rn with no assumptions on the sign of f. Here we state this theorem together with some related results and we outline the main features of the problem.